# Killing time and people softly…

What would happen if you put a person in a microwave?

As a way of whiling my Sunday morning, I decided to ‘solve’ this unique question on Quora. The question was: What would happen if you put a person in a microwave? Yes, Quora has its dark side, and I’m loving it! Below is the answer.

Firstly it’s a terrible logistical and ethical problem. Considering that we have pushed these aside, we need to start by assuming that the person is a large sphere of some radius a! (Yep, physicists love to approximate spheres!)

Some standard mathematical approximations (Skip to the last paragraph if only interested in the final result):

We need to find the temperature on the surface as a function of time. The human is initially at temperature $T_0$

Now after a few seconds, the temperature boundary conditions are:

$T(0,0)=T_1$

$T(a,0)=T_0$

This is because a microwave heats mainly from the inside. Unlike an oven, the microwave will first heat up the centre and this heat then diffuses throughout the rest of body. (You might have probably noticed that while heating something in the microwave. Even if the outside is at room temperature, the insides are piping!) Now both $T_0$ and $T_1$ are functions of time.

$T_1$, the temperature at the centre directly depends on the configurations of the microwave, while $T_0$ will depend on the thermal diffusion rate from the centre to the surface.

Calculating $T_1$ is dodgy and depends on accounting for heat loss by evaporation rates of water and then extrapolating. However, for our timescales, we can consider $T_1$ to be more or less constant.

Now, the thermal diffusion equation for a sphere in steady state gives us: (T being the temperature)

$\nabla^2 T=0$

This should give us a general solution of the form:

$T=A+B/r$

with $r$ as the radial coordinate.

Thus, inspired by the steady state solution, we can write:

$T(r,t)=T_0+B(r,t)/r$

Thus $B(r,t)$ can be written as $r(T-T_0)$

This gives us:

$\frac {dB}{dt}=D \frac{\partial^2 B} {\partial r^2}$

where $D= \frac{\kappa}{C}$

I’m composed of quite a decent proportion of laziness (about $80\%$) so am skipping a few steps which mean that I would not need to type out several lines of equations in $\LaTeX$. It should suffice to say that here I am merely converting the thermal diffusion equation to the standard 1-D case which is easier to solve.

This gives us $B(0,t)=B(a,t)=0$, the 2 boundary conditions. Also  $B(r,0)=r(T_1-T_0)$, since $T=T_1$ at $t=0$. (ie the temperature at the centre due to microwave)

And feeding back these equations back to our diffusion equation, we obtain a solution of the kind given below. The general solution will involve expanding these terms with some coefficients.

$B_n=\sin (n\pi r/a) e^{-D(n\pi/a)^2 t}$

$B(r,t)=\Sigma_{n=1}^{\infty} A_n \sin (n\pi r/a) e^{-D(n\pi/a)^2 t}$

Similarly, we can obtain the coefficients $A_n$. Actually it involves wrting out the expansion with $A_n$ for $t=0$ and using the orthogonality condition. $A_n$ comes to $\frac {2a}{n\pi}(T_1-T_0)(-1)^{(n+1)}$

Combining both $A_n$ and $B_n$, we finally obtain for the surface of the human body, temperature as:

$T(a,t)=T_0+\frac {2a}{\pi}(T_1-T_0)\Sigma_{n=1}^{\infty} \frac{(-1)^{(n+1)}}{n}\sin (n\pi) e^{-D(n\pi/a)^2 t}$

So, now let’s plug in some values.$T_0,$ the average body temperature is $37^{\circ}$C.

$D$, the thermal diffusivity is given by the ratio of conductivity to specific heat capacity. (It’s actually the Heat Capacity per unit volume for pedants. But since humans are mostly water, weight and volume cancel out.) Guiltily browsing figures for human thermal conductivity and heat capacities, I jotted down some figures. Again, plugging in those values, we get $D=0.543/3470=1.6\times 10^{-4}$.

I estimated the average chest width, $a$ to be 1 m from available figures.

Now putting them back:

$T(1,t)= T_0+ 0.63(T_1-T_0)\Sigma_{n=1}^{\infty} \frac{(-1)^{(n+1)}}{n}\sin (n\pi ) e^{-1.6\times 10^{-4}(n\pi)^2 t}$

Conclusion:

The exponential term is extremely small. The second term only starts to matter heavily when $t \approx 10^3$ seconds or about 17 minutes or greater, which means that it takes at least 1/4th of an hour for the temperature at the skin to reflect significant changes. Thus, under approximations made, it should take more than 17 minutes to completely cook a human alive for temperatures sufficiently greater than $37^{\circ}$C independent of configurations of the microwave.

Thus we see that the time rate to fry the human is mainly dependent on $T_1$ and thus the rate at which the microwave heats the centre of the human. Although other effects like surface currents due to the varying electric and magnetic field apart from an intense burning at the centre might not be a pleasant experience as well have not been considered, these could play important roles as well in the heating. Else, he would slowly be evaporated from inside out as his body is drained and heated at the same time. The human body is about $80\%$ water and what will be left of him in the microwave will probably be a mess best left for the morgue!

Caveat:

I would need to add if this wasn’t apparent already is that this is an order of magnitude estimate. Microwaves don’t really heat from the centre outwards, but it should give a reasonable enough estimate all the same. A better way to look at the problem including any further mathematical considerations that may be considered are welcome from anyone who has chanced upon this crazy article!